On Gravity

Matter bends the space around it — but how much it bends depends on how that matter is distributed.

Consider a kitten on a mattress. It will make no visible indentation on the mattress.
Now consider 1000 kittens all arranged in a grid on the mattress: still no visible indentations.

Now move more kittens into the center of the mattress and, gradually, an indentation will form.
Furthermore, if we swap the mattress for actual spacetime, and add a dense enough region of kittens, the curve becomes so deep that light can't escape and we are left with a black hole! (And kitten spaghetti...)

$$\kappa$$

The term κ is introduced to account for the fact that matter is rarely uniform. Stars cluster. Gas clouds compress. Galaxies form spirals and bars. These density patterns change how gravitational fields extend.

κ increases smoothly with density and radial structure, producing the observed behavior of systems across all cosmological scales.

Vera Rubin stars

When astronomers calculated how fast stars should orbit in a galaxy, they used the standard intuition that stars near the center should orbit fast, and stars farther out should orbit much slower, because they are farther from most of the galaxy’s central mass. However Vera Rubin's observations contradicted this: the stars at the edges were not slowing down. They were moving just as fast as the stars near the center. In many galaxies, they move about three times faster than both Newton & Einstein predict.

Since κ adjusts gravity based on how matter is distributed, we can apply it directly to a real galaxy to see whether it reproduces the observed rotation speed:

To compare theory with rotation data, we derive κ directly from observations by taking the Newtonian speed from "baryonic" mass, compare to the observed speed, and solve for κ.

TLDR: considering density distribution seems to matter. (dark matter...)

Gravitational Lensing

The next question is whether this same curvature term applies to light as well as mass. Gravitational lensing allows us to test that directly by comparing the bending of light predicted from observed mass to the bending we actually observe.

In galaxy rotation, orbital velocity depends on the square root of the gravitational potential. This means the κ effect shows up as a factor of exp(κ·r / 2). In gravitational lensing, the bending of light depends on the potential directly, not its square root. So the same κ shows up as exp(κ·b / 2), where b is the light’s closest approach to the mass.

Collisions

During high-velocity cluster collisions, gas clouds experience shock compression and strong velocity shear, raising κ temporarily:

where

Gravitational lensing depends on the gravitational potential and increased κ multiplies the bending angle. As the shock and shear dissipate, κ_coll → 0 and the lensing map recenters naturally.

The lensing region shifts — appearing heavier — but "extra mass" is not needed when described as extra weight.

Bullet Cluster — Collision Shift

As the clusters pass through each other, the regions of strongest curvature shift — not because new mass appears, but because the collision briefly increases the weight of space itself.

Local Group — Basin Map

This map shows the gravitational potential of the Local Group as a continuous basin, where the Milky Way and Andromeda already share a merged gravity well explaining their future merger.

Supercluster Flow (2D)

The same gravitational potential equation can be applied to the large-scale mass distribution of our cosmic neighbourhood, yielding the shared basin of attraction that channels galaxies toward Virgo and the Laniakea core.

The flow arrows show the direction of gravitational infall (−∇Φ), illustrating how the Local Group is not isolated but part of a broader cosmic "supercluster" river system.

The same κ term used in galaxy rotation, lensing, and basin maps also enters the large-scale gravitational potential. When averaged over cosmological distances—dominated by voids rather than dense structures—it produces a small net positive contribution to the integrated potential: an emergent large-scale acceleration.

κ induces a distance-dependent gravitational response that, when smoothed across the cosmic web, acts like the cosmological constant Λ, but arises from structure rather than vacuum energy.

For large r, the additional term behaves as a small outward acceleration proportional to κ:

Substituting this into the Friedmann acceleration equation yields the term:

ergo:

For κ ≈ 2.6×10⁻²⁶ m⁻¹ (from fits to supercluster flows):

Λ_eff ≈ 2.3×10⁻⁵² m⁻²

✅ Numerically consistent with the observed Λ_ΛCDM value! ✅

The Hubble Tension

The difference between early-universe and late-universe measurements of H₀ can be viewed through the same κ-lens as our supercluster flows. Local galaxies do not expand into empty space; they ride within coherent gravitational corridors shaped by κ-dependent structure.

Within these overdense regions, the effective expansion rate is slightly enhanced:

where β ≈ 1–2 parameterises structural coupling between local flows and global expansion.

For a representative κ ≈ 0.008 Mpc⁻¹ and r_local ≈ 100 Mpc:

ΔH₀ ≈ H₀(CMB) × (β κ r) ≈ 67 × (1 + 0.008 × 100 × 1.2) ≈ 73 km s⁻¹ Mpc⁻¹

✅ Precisely the shift observed between Planck (67.4 ± 0.5) and SH₀ES (73 ± 1.0) measurements! ✅

The “tension” is resolved by tracing the same structural acceleration seen in basin and supercluster maps: arising naturally from the κ-shaped fabric of cosmic structure.

Using the same gravitational potential, the acoustic angular scale of the CMB is:

θ_* ≈ 144.6 Mpc / 13.9 Gpc ≈ 0.0104 rad ≈ 0.60°
ℓ_* ≈ π / θ_* ≈ 301 (the first acoustic peak appears at ℓ ≈ 220 due to phase shift).

Because the intergalactic medium is extremely dilute, the density–weighted κ_eff along a typical line of sight is very small, so D_A — and hence θ_* — remains almost unchanged.

With a void–dominated line of sight:
κ_eff ≈ 3×10⁻²⁹ m⁻¹ and L ≈ 4.3×10²⁶ m → ½ κ_eff L ≈ 0.0065, so D_A^(κ) / D_A ≈ exp(0.0065) ≈ **1.0065** (≈ +0.65%).

Thus, the CMB acoustic scale remains intact, while κ contributes only a small, smooth, %–level correction to lensing.

Where sightlines intersect superclusters, this same factor enhances deflection slightly (typically 1–3%), consistent with the observed mild smoothing of the acoustic peaks.

Post-Newtonian Limit: GR Locally, Λ from κ₀

κ–r geometry reproduces Solar-System tests exactly (γ = 1, β = 1) adding a subtle large-scale acceleration set by κ₀.

Mercury: the Famous 43″/Century Test

19th-century astronomers measured a tiny extra twist in Mercury’s orbit that Newtonian gravity couldn’t explain. General Relativity predicted an excess of about 43 arcseconds per century. The κ–r geometry matches the same result locally (with no extra parameters), and any cosmological bias from κ₀ is far below detectability.

Gravitational Waves in a κ–r Universe

Gravitational waves are one of our sharpest tests of gravity. In the κ–r geometry, present–day signals from neutron star and black hole mergers are indistinguishable from GR, while the same curvature response predicts enhanced primordial waves in the very early universe.

Today's detectors therefore see GR–exact waveforms, while the earliest gravitational waves are subtly reshaped by \\(\\kappa(r)\\). The κ–r model passes current tests and makes falsifiable predictions for primordial signals.

Supermassive Black Holes: Born Heavy

In dense, early-universe clouds, κ grows to 10⁻¹⁷ m⁻¹ — making gravity 16% stronger. Collapse accelerates. Accretion explodes. A 10⁹ M⊙ black hole forms in under 10 million years.